Properties

Label 1-4033-4033.57-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0494 - 0.998i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.286 + 0.957i)3-s + (−0.939 + 0.342i)4-s + (−0.549 + 0.835i)5-s + (0.893 − 0.448i)6-s + (−0.0581 + 0.998i)7-s + (0.5 + 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (−0.597 − 0.802i)12-s + (−0.893 + 0.448i)13-s + (0.993 − 0.116i)14-s + (−0.957 − 0.286i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.286 + 0.957i)3-s + (−0.939 + 0.342i)4-s + (−0.549 + 0.835i)5-s + (0.893 − 0.448i)6-s + (−0.0581 + 0.998i)7-s + (0.5 + 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (−0.597 − 0.802i)12-s + (−0.893 + 0.448i)13-s + (0.993 − 0.116i)14-s + (−0.957 − 0.286i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0494 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0494 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0494 - 0.998i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0494 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1790762237 + 0.1704240706i\)
\(L(\frac12)\) \(\approx\) \(-0.1790762237 + 0.1704240706i\)
\(L(1)\) \(\approx\) \(0.5587100274 + 0.2374514893i\)
\(L(1)\) \(\approx\) \(0.5587100274 + 0.2374514893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (-0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.918 + 0.396i)T \)
13 \( 1 + (-0.893 + 0.448i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 - T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (-0.802 + 0.597i)T \)
59 \( 1 + (0.286 - 0.957i)T \)
61 \( 1 + (0.802 - 0.597i)T \)
67 \( 1 + (-0.918 + 0.396i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (0.993 + 0.116i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (0.957 - 0.286i)T \)
97 \( 1 + (0.727 + 0.686i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.74787437680877444227377304069, −17.324249507817629659059669887829, −16.45886422566224975443744202036, −16.227791904947487366252343418802, −15.17518922973847881912774519189, −14.55509123767045755426070728482, −13.88944743647032214134669905840, −13.1399986934944127800825291379, −12.78540761772733277513660607882, −12.114218011820281267425841380748, −10.94663078712329975100043021700, −10.30459224720316370013632582592, −9.2597268803363029137233253365, −8.69732984597058117084121349662, −7.95575156421928528489700237013, −7.47036271673689828553537840025, −7.02405919731333313420985899232, −6.07737777060988207560789428814, −5.189777336516583888711813695954, −4.68428562451199676897769294357, −3.7098511126237707911733864780, −2.82295058310301971863853216764, −1.59910328096874879261611656359, −0.56411874981821179966431503374, −0.12007892052391922727639310620, 2.01671020127505082763900303721, 2.347558477112698545509147551109, 3.14813256424545577842326343451, 3.86018928797427706762882138020, 4.54267572561700990389162714655, 5.328675933774441667289871044145, 6.10021240294903447953558332866, 7.44111623608730853937760793381, 8.06283473850862053585555868372, 8.67481578971872090037468602848, 9.73183581600431148000092820413, 9.8150115403632265646559975263, 10.83132715396842341948282884831, 11.2474999646026104646631567529, 11.92893946504236392320141072926, 12.73369806764558826038314041427, 13.36285405297522019762510352446, 14.5384066734461388729982452879, 14.823235273524643459454265683173, 15.37757470073987345449034445785, 16.18905983612310166378026929401, 17.15996740265537165468451621132, 17.61758779538013602157499303321, 18.78520864663913993939452811637, 18.9038936599192024756117940181

Graph of the $Z$-function along the critical line